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Trigonometric Equations

How many solutions lie in an interval. Reduce to one ratio of one angle, then count the roots on the interval itself.

Questions in the bank
51
q/paper in 2025–26
0.32
Numeric answer
33%
Notes pages
4

Tier: Long tail

When you’ll see it

The number of solutions of a trigonometric equation in a given interval, or the values of a constant for which a solution exists.

How this chapter is tested

Most questions ask only how many solutions lie in an interval, so the general solution is a tool, not the answer. The usual route is to reduce everything to one ratio of one angle — by a quadratic in sin θ or cos θ, a product-to-sum step, or a multiple-angle identity — solve for the ratio, drop values outside [−1, 1], and count angles period by period.

The count is where marks are lost. A value of ±1 is reached once per period, not twice; a closed interval can hold a root at each end; two families of roots can overlap; and collapsing to tan 3x can admit roots where an original tan x is undefined. Marking each root on the interval is safer than trusting a formula.

The rest turns on ranges: f(x) = k has a solution only when k lies in the range of f, and a cos x + b sin x = c needs c² ≤ a² + b². Some equations are settled only by bounds or a graph. The identities come from Trigonometric Identities, and a determinant in θ set to zero in Determinants ends here.

The sub-skills

The distinct skills inside the chapter, in the order to learn them.

  • Quadratic in one ratio

    Solve for sin θ, cos θ, sec θ or cosec θ, discard roots outside the range, then count the angles in the interval.

  • Product-to-sum and multiple angles

    Turn products into sums to reach cos A = cos B or a product equal to zero, or collapse to one ratio of 3x or 4x.

  • Range and existence of solutions

    f(x) = k is solvable exactly when k is in the range of f; write a cos x + b sin x as √(a² + b²) cos(x − α).

  • Exponential, bounded and graphical equations

    For a > 1, a to the power sin²x lies in [1, a]; sides that meet only at their bounds; a line against the graph of tan x.

Traps to expect

Distractor shapes this chapter reuses. The Traps page covers the ones that cut across chapters.

  • ±1 and the endpoints

    cos θ = −1 has one root per period, and on [−π, π] it holds at both ends.

  • Cancelling a ratio

    In sin θ cos θ = sin θ, cancelling sin θ loses θ = π. Factor as sin θ(cos θ − 1) = 0.

  • Overlapping families

    Two families of roots can share some angles; adding the two counts without removing the shared ones overcounts.

  • Squaring adds roots

    Squaring a cos x = c − b sin x also admits roots of a cos x = −(c − b sin x); check each root, or use the auxiliary angle.

Learn it before you drill it

This chapter has full teaching notes — foundations, worked examples, self-checks and a mastery check for each page. Read the notes once, then drill page by page below.

Trigonometric Equations notes

Drill every Trigonometric Equations question

51 questions from the bank, across 4 subtopics.

Drill the 51 questions

Drill one subtopic at a time

The 4 subtopics, in teaching order.

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